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Discussion Papers
The Political Economy of Sin Taxes
Markus Haavio
Bank of Finland and HECER and
Kaisa Kotakorpi University of Tampere
Discussion Paper No. 185 September 2007 ISSN 1795-0562
HECER – Helsinki Center of Economic Research, P.O. Box 17 (Arkadiankatu 7), FI-00014 University of Helsinki, FINLAND, Tel +358-9-191-28780, Fax +358-9-191-28781,
HECER
Discussion Paper No. 185
The Political Economy of Sin Taxes*
Abstract
We analyse the determination of taxes on harmful goods when consumers have self- control problems. We show that under mild conditions, the socially optimal tax rate exceeds the average distortion caused by self-control problems. Further, we show that in most cases the tax rate chosen in political equilibrium is below the socially optimal level.
JEL Classification: H21, H30, D72.
Keywords: Excise taxation, voting, self-control.
Markus Haavio Kaisa Kotakorpi
Bank of Finland Department of Economics and Accounting
P.O. Box 160 University of Tampere
FI-00101 Helsinki FI-33014 University of Tampere
FINLAND FINLAND
e-mail:
markus.haavio@bof.fie-mail:
kaisa.kotakorpi@uta.fi* We would like to thank Essi Eerola, Jukka Pirttilä, Panu Poutvaara and Theis Theisen as
well as seminar participants at the 1st World Meeting of the Public Choice Society, the
63rd Congress of the International Institute of Public Finance, and at HECER for
comments. This work was initiated while Kotakorpi was visiting the LSE. Kotakorpi thanks
the LSE and STICERD for their hospitality, as well as Tim Besley and Erik Eyster for
helpful comments on an early version of the paper. All remaining errors are ours. Kotakorpi
gratefully acknowledges financial support from the FDPE, the European Commission and
the EDNET network, Yrjö Jahnsson Foundation and the Foundation for Economic
Education.
1 Introduction
We analyse the determination of taxes on goods whose current consumption causes util- ity costs (for example negative health e¤ects) in the future. When consumers have time- inconsistent preferences, they consume too much of such goods. Using “sin taxes” to correct distortions in the consumption of harmful goods when consumers have self-control problems has also been considered in O’Donoghue and Rabin (2003; 2006).
Market-based mechanisms for correcting the distortion caused by self-control problems are likely to be ine¤ective (see Köszegi (2005)), and consumers might thus value sin taxes as a commitment device. In addition to the monetary cost of taxation, sin taxes a¤ect individual utility due to the corrective nature of the tax when preferences are time-inconsistent. If this positive e¤ect outweighs the monetary cost, sin taxes can improve individual welfare - see Gruber and Köszegi (2004) and Kotakorpi (2006) for theoretical analyses and Gruber and Mullainathan (2005) for empirical evidence.
The purpose of the current paper is two-fold. Firstly, we contribute to the discussion on optimal sin taxes by deriving an explicit rule for the optimal tax in a second-best setting where individuals di¤er in their degree of self-control problems. Secondly, we examine how sin taxes are determined in political equilibrium, a question that has to our knowledge not yet been analysed in the literature.
Many economists have been concerned that sin taxes as well as other paternalistic poli- cies that are aimed at helping irrational individuals1 are often detrimental for the welfare of rational individuals.2 This has resulted in an emphasis on the search for policies that help irrational individuals while having only a small or no impact on those who are rational.3 However, there has recently been an interest in moving beyond studying minimal interven- tions, to studying optimal paternalistic policies. Our analysis is particularly closely related in this respect to O’Donoghue and Rabin (2006).4
When individuals di¤er in their degree of self-control problems but a uniform sin tax is applied, we are necessarily in a second-best situation. We analyse the trade-o¤ between bene…ts to irrational individuals and costs to rational individuals further, and …nd the optimal balance between them: we provide an explicit formula for the second-best optimal sin tax
1Throughout the paper, we refer to individuals with a self-control problem as irrational, as they behave in an inconsistent manner and make consumption decisions that fail to maximise their own life-time utility.
Similar terminology has been used for example by O’Donoghue and Rabin (2006).
2Beyond this concern, some economists remain sceptical about paternalism for more general reasons - see for example Glaeser (2006) for a critical view. For example, the possibility of government failure may reduce the e¤ectiveness and desirability of paternalistic policies. Despite the importance of this consideration, we abstract from this issue in the current paper. On the other hand, we show that in our model consumers would themselves vote for paternalistic policies: such policies can therefore be the outcome of a democratic decision making process, which has interesting implications for the justi…cation of paternalism.
3See for example Camereret al (2003), Thaler and Sunstein (2003) and O’Donoghue and Rabin (1999).
4O’Donoghue and Rabin (2006) examine the conditions under which the optimal sin tax is positive, and whether sin taxes can yield Pareto improvements (compared to zero taxes). They further provide comparative statics of the optimal sin tax when there are changes in the distribution of self-control problems and tastes.
and show that it exceeds the mean distortion in consumption. The reason is right at the heart of the recent discussion on paternalism: for reasonable assumptions about the form of the demand function, sin taxes have a relatively small (negative) e¤ect on the utility of (close to) rational individuals, who consume relatively little of the good. On the other hand, irrational consumers with a very high level of consumption gain a lot from sin taxes.
We then turn to analyse the majority voting equilibrium when individuals di¤er in their degree of self-control problems. We assume that individuals are fully aware of their self-control problem, and vote on the sin tax to be implemented from the next period onwards. Taxation can then provide a commitment device that helps individuals move their consumption closer to its optimal level.
As a benchmark, we consider the case where tax revenue is distributed back to con- sumers in such a way that the redistributive e¤ects of taxation are eliminated. In this setting individuals prefer the level of taxes that completely eliminates the distortion in their own consumption, and the political equilibrium is the tax rate that corresponds to the median level of self-control problems. We show that equilibrium taxes are in most cases below the socially optimal level. The reason is that the asymmetric e¤ect of sin taxes at di¤erent ends of the distribution of self-control problems is not taken into account by the median voter.
However, in this setting where sin taxes have no redistributive e¤ects, there is one particular case where the equilibrium and the social optimum coincide: this is when consumption is so harmful that the optimal level of consumption is zero even in the absence of taxation. In this case, it is in the interests of both the consumers and the social planner to eliminate all consumption.
We then proceed to analyse the case where the redistributive e¤ects of sin taxes are taken into account. This case is the one that is more relevant from a practical point of view, and also highlights some interesting new issues. On the one hand, an individual without self-control problems will prefer a low tax, as high taxation would cause an unnecessary distortion in consumption from his point of view. On the other hand, however, sin taxes will redistribute income from irrational large-scale consumers to rational consumers, a reason for consumers with no self-control problems to vote for a high tax.
Despite these counteracting motives that a¤ect an individual’s preferred tax rate, we show that a majority voting equilibrium exists in our model also in this case, and corresponds to the tax rate preferred by the individual with the median level of self-control problems.
Importantly, we show that when redistributive e¤ects of sin taxes are taken into account, the equilibrium tax rate is below the socially optimal levelregardless of the level of harm from consumption. The redistributive motive for taxation implies that equilibrium taxes are below the social optimum even when consumption is extremely harmful: the median voter does not consume the good in equilibrium, and simply wants to maximise redistribution from irrational individuals towards himself. On the other hand, the social planner wants to completely eliminate consumption. It is worth emphasising that our results regarding extremely harmful
consumption goods are very robust to di¤erent functional form assumptions and do not depend on the distribution of self-control problems.
One of the aims of our analysis is to contribute to the policy discussion on the taxation of harmful goods. In European countries, tobacco products are taxed much more heavily than alcohol: the excise duty on the most popular brand of tobacco was on average approximately 60 % of the total retail price in the EU-15 member states in 2003 (Cnossen and Smart 2005), whereas the corresponding …gure was 19 % for beer, 14 % for wine and 39 % for spirits (WHO 2004).5 It might appear that cigarette taxes are too high from a social point of view, particularly as cigarette taxes in most countries seem to exceed the external costs of smoking (Cnossen and Smart 2005)6. However, considering not only negative externalities, but also harm experienced but not taken into account by the consumer himself, optimal taxes should indeed exceed the level that would be appropriate if only externalities were taken into account.
As our analysis shows, the redistributive motive for taxation implies that equilibrium taxes on highly harmful goods such as cigarettes may be too low from a social point of view.7
In addition to the previous literature on taxation when consumers have self-control prob- lems, our analysis has some similarities with the analysis of commodity taxation in the presence of externalities: negative health e¤ects (in the case of consumers with self-control problems) as well as negative externalities are both harmful e¤ects not taken into account by consumers, and governments might wish to alleviate these e¤ects through taxation. Diamond (1973) has analysed optimal taxation of externality-generating goods when individuals give rise to di¤erent (marginal) externalities.8 In the case of externalities, however, there is no natural assumption to make about how the magnitude of the marginal externality should be correlated with individual demand. In our context, on the other hand, a high marginal internality is naturally associated with high consumption, since consumers with a more severe self-control problem have a higher level of consumption, ceteris paribus. This correlation is the mechanism that drives many of the key results in this paper.
The rest of the paper proceeds as follows. We introduce the model in Section 2 and derive the second-best optimal sin tax in Section 3. The political equilibrium is analysed in Section 4, where we …rst study the benchmark case where sin taxes have no redistributive e¤ects, and then extend the analysis to account also for the redistributive e¤ects of sin taxes. Section 5 shows that our main conclusions extend to the case where sin taxes are also used for revenue
5The …gures for alcoholic drinks were calculated using data for 9 countries only, as the …gures for the rest of the EU-15 were not reported (see WHO 2004, 54). The average total tax collections in the EU-15 member states were approximately 100 euros per adult in the case of alcohol, and around 280 euros per capita in the case of tobacco (see for example Cnossen (2006a) and (2006b)).
6In the case of alcohol, on the other hand, taxes appear to be lower than the level that would be mandated even by externality considerations alone (Cnossen 2006a)
7The relatively low prevalence of smoking suggests that cigarettes …t our category of highly harmful sub- stances (where most people abstain from consumption): smoking prevalence is around 20-30% in most EU countries, whereas only around 15% of adult Europeans abstain from alcohol consumption (Anderson and Baumberg 2006, European Commission 2004).
8See Eerola and Huhtala (2007) for a recent contribution to the literature on the political economy aspects of environmental policy.
raising purposes. Section 6 concludes.
2 The model
We consider a model where consumers have a quasi-hyperbolic discount function (Laibson 1997), using a set-up that is similar to O’Donoghue and Rabin (2003; 2006). Life-time utility of an individual is given by
Ut= (ut; :::; uT) =ut+ i XT s=t+1
s tus; (1)
where i; 2(0;1)andutis the periodic utility function. Individuals are therefore assumed to be identical in all other respects, but they di¤er in their degree of quasi-hyperbolic discount- ing. We assume that the quasi-hyperbolic discount factor has a distribution function F( ) with meanE( )and median med:Throughout the paper we consider the general case where has the support [ L; H], with 0 L < H 1. Quasi-hyperbolic discounting implies that preferences are time-inconsistent: discounting is heavier between today and tomorrow, than any two periods that are both in the future.
We assume that individuals derive utility from a composite good (z), which is taken as the numeraire, and another good (x) which is harmful in the sense that it yields positive utility in the short-run, but has some negative e¤ects in the long-run. Speci…cally, we assume that periodic utility is given by
ut(xt; xt 1; zt) =v(xt) h(xt 1) +zt; (2) wherev0 >0; v00 <0 and the harm function9 is characterised byh0 >0and h00>0:
We assume that there is no borrowing or lending. In each period, consumers then choose xto maximize10 u(x) =v(x) i h(x) +zsubject to a per-period budget constraintqx+z B+S. We assume that product markets are competitive and normalise the producer price to 1, andq = 1 + denotes the consumer price of goodx: B is the consumer’s income (taken to be exogenous) andS is a possible lump-sum subsidy received by the consumer from the government. Taxes and subsidies will be modelled in more detail in later sections. Given the above speci…cation, the demand for goodx satis…es
v0(x ) i h0(x ) =q: (3)
9As in O’Donoghue and Rabin (2006), we assume that the marginal bene…ts and marginal costs of con- sumption are independent of past consumption levels. In such a setting, it is not essential that the harm is modelled as occuring only in the period following consumption -hcan be thought of as the discounted sum of harm occurring in all future periods. See Gruber and Köszegi (2004) for an analysis where past consumption a¤ects current marginal utility.
1 0We have dropped the time indext, since with our speci…cation consumption is constant accross periods.
However, the time-inconsistency in preferences implies that the consumer would like to change his behaviour in the future: from the next period onwards, the consumer would like to choose consumption levels that maximise11 uo(x) =v(x) h(x) +z: We take this long-run perspective as the one relevant for welfare evaluation - this has become a standard choice in the literature on sin taxes (see for example Gruber and Köszegi (2004), O’Donoghue and Rabin (2003; 2006)). There are clear reasons that justify this choice of welfare criterion: Firstly, we assume that taxes are implemented from the period after the policy decision is made.
Therefore, consumers themselves agree thatuo(x) is the relevant utility function, and voting decisions will be made based on maximising this function. We thus use the same criterion consistently when deriving both the optimum and the equilibrium level of taxes. Further, uo(x) is the utility function that applies to all periods except for the present one. Since we consider an in…nite number of periods, the weight of any single period should be negligible as long as periods are su¢ ciently short12. This latter consideration applies irrespective of the timing of the model.
The optimal level of consumption therefore satis…es v0(xo) h0(xo) = q : because of quasi-hyperbolic discounting ( < 1); the equilibrium level of consumption of the harmful good(x ) is higher than the optimal level(xo):
3 The second-best optimal sin tax
We argued above that long-run utility is the appropriate welfare criterion in our model. The social welfare function is then given by W (q) R H
l G(V (q; ))f( )d , where V (q; ) is the long-run indirect utility function. We assume that the functionG(:) is utilitarian, and the social welfare function therefore becomes
W (q) =R H
l V (q; )f( )d =E [V (q; )]
=E [v(x (q; )) h(x (q; )) qx (q; ) +S(q; )] +B;
wherex satis…es (3) and is therefore distorted whenever <1, as argued above. Taking into account the government’s budget constraint E [x (q; )] =E [S(q; )], the social welfare function can be written as
W (q) =E [V (q; )] =E [v(x (q; )) h(x (q; )) x (q; )] +B:
Given the distortion in consumption caused by quasi-hyperbolic discounting, the govern- ment may consider imposing a sin tax on harmful goods as a corrective measure. The social
1 1See equation (1) and think of a consumer in period t, making consumption decisions for period t+ 1 onwards.
1 2See also Bernheim and Rangel (2007, 4) for a discussion on this point.
planner’s …rst-order condition is E h@V(q; )
@q
i
=E h
[v0(x (q; )) h0(x (q; )) 1]@x@q(q; )i
=E h
[ (1 ) h0(x (q; ))]dxdq(q; )i
= 0; (4)
where the last step was obtained by using (3).
As we consider a case where a uniform tax is applied, choosing the optimal tax involves a trade-o¤ between helping consumers with a severe self-control problem, whilst causing a distortion for those who are rational. From (4), the second-best optimal tax is given by
o= E (1 )h0(x (q; )) +
Cov h
(1 )h0(x (q; ));@x@q(q; )i E h@x (q; )
@q
i : (5)
It should be noted that the socially optimal tax rate is independent of the way in which tax revenue is distributed back to consumers, that is, of the form of the function S( ; ):Therefore, in both of the cases considered below - that is, regardless of whether sin taxes have redistributive e¤ects or not - the socially optimal tax rate is given by (5).
In order to get clearer results in the current and the next section, we make the following assumption about the functional forms ofv(x) and h(x):
Assumption 1 (i) v000(x) 0 and (ii) 2 h000(x)h0(x)
[h00(x)]2 1:
Assumption 1 is satis…ed for commonly used functional forms, for example whenvis of the CRRA or CARA-variety13 or quadratic, and when the harm function is linear14, quadratic, exponential orh(x) =xs wheres 4=3.
We can now proceed to analyse the socially optimal tax rate in (5). The …rst term, E[(1 )h0(x )];is the average distortion caused by self-control problems in the economy.
It can be shown that the second term in (5) is positive given Assumption 1. We can therefore state the following proposition:
Proposition 1 The socially optimal sin tax is higher than the average distortion caused by self-control problems, that is, o> E [(1 )h0(x (q; ))].
Proof. See the appendix.
It is shown in the appendix that a su¢ cient condition for Proposition 1 is that @2x@q@(q; ) >
0, which holds given Assumption 1. The condition @2@q@x (q; ) > 0 has a very intuitive expla- nation: it implies that taxation has a larger impact on irrational consumers with a very high
1 3Kimball (1990) provides an economic interpretation of the conditionv000>0;albeit from a context that is rather di¤erent from ours: v000>0is associated with the concept of prudence, and implies that precautionary savings of risk averse individuals increase with increased uncertainty.
1 4Note that when the harm function is linear, part (i) of Assumption 1 has to hold as a strict inequality.
level of consumption( 0)than on rational consumers with a moderate level of consump- tion( 1). The bene…t of a high sin tax for consumers with a severe self-control problem thus exceeds the (negative) impact on the utility of (close to) rational individuals, and the socially optimal sin tax therefore exceeds the average distortion.
4 Political equilibrium
From the point of view of the consumer, the problem is that he would like to consume less in the future, but repeatedly fails to do so due to self-control problems. We assume throughout the analysis that consumers are sophisticated - that is, they are completely aware of their self- control problem15. However, market-based mechanisms for correcting the distortion caused by time-inconsistent preferences are likely to be ine¤ective (see Köszegi (2005)): even though both consumers and …rms would have the incentive, ex ante, to sign contracts that implement the optimal level of consumption, in a competitive market consumers cannot be prevented from purchasing from other …rms ex post. Consumption of harmful goods is therefore as if only a spot market was available (that is, suboptimally high). To the extent that laws on commodity taxation cannot be changed each period, sophisticated consumers might thus value sin taxes as a way of committing to a lower level of consumption in the future.
In this section we analyse the level of taxes that will emerge in a political equilibrium and compare the equilibrium tax rate with the social optimum. In each case considered below, we assume that consumers vote over a sin tax to be implemented in all subsequent periods, starting from the period following the vote16. As the utility from all subsequent periods is discounted exponentially, the individual’s policy preference function is given by his indirect long-run utility function,V (q; i) =v(x (q; i)) h(x (q; i))+z: the individual’s preferred tax rate will be the one that maximises his long-run utility, taking into account the fact that actual consumption decisions will be distorted in the absence of a sin tax.
Finally, we assume that the outcome of the vote is determined by direct majority rule17.
1 5The concepts of sophistication and naivete (complete unawareness of ones’ self-control problem), were discussed already by Strotz (1955-6) and Pollak (1968) and have been recently analysed in numerous papers - see for example O’Donoghue and Rabin (1999) for an analysis of the implications of both sophistication and naivete, and O’Donoghue and Rabin (2001) for a model that introduces a formalisation of the intermediate case of partial naivete. Since there are no intertemporal linkages in the marginal bene…ts and costs of con- sumption in our model, consumption decisions (in the absence of commitment) would be the same for naifs and sophisticates. However, voting decisions depend on whether the individual is aware of his self-control problem: (partially) naive individuals would vote for a lower tax than sophisticated individuals.
1 6If consumers were to vote on taxes only for this period, all consumers would vote for zero taxes; and if they were to vote on taxes to be implemented forever butincludingthe current period, they would vote for a lower level of taxes than implied by the analysis below (the socially optimal level of taxes would also be lower;
see Gruber and Köszegi (2004, 1967)) .
1 7The same results hold if there is a representative democracy with two-party electoral competition, the parties can fully commit to a tax policy and care only about their chances of being elected (and do not have preferences over the level of taxes themselves) - this would be a simple case of Downsian electoral competition (Downs (1957); see also Persson and Tabellini (2000)).
4.1 Benchmark: sin taxes have no redistributive e¤ects
Even though the social optimum is una¤ected by the way in which tax revenue is distributed back to consumers, consumers clearly will not be indi¤erent about the subsidies that they receive. The shape of the functionS( ; )thus has an e¤ect on the political equilibrium. Con- sider …rst the case where the tax has no redistributive e¤ects18, namelyS( ; ) = x (q; ):19 The long-run indirect utility function of individualiand therefore the policy preference func- tion of individualiis then given by
V (q; i) = v(x (q; i)) h(x (q; i)) (1 + )x (q; i) + x (q; i) +B (6)
= v(x (q; i)) h(x (q; i)) x (q; i) +B:
wherex again satis…es (3).
4.1.1 The case of moderately harmful consumption
In the case where sin taxes have no redistributive e¤ects, the comparison between the equilib- rium and the social optimum turns out to depend on the extent of harm from consumption.
Let us …rst analyse the political equilibrium in the case where current consumption causes harm in the future, but the optimal (rational) level of consumption is nevertheless positive at zero taxes, that is v0(0) h0(0) 1>0. Using similar steps as in the previous section, the …rst-order condition is given by
@V (q; )
@q = (1 ) h0(x (q; )) @x (q; )
@q (7)
and each individual’s preferred tax rate is given by
( ) = (1 ) h0(x (q; )): (8)
The policy preference function (6) is single-peaked20, and a majority voting equilibrium therefore exists and the tax rate preferred by the voter with the median most preferred level of taxes is chosen in equilibrium. Further, policy preferences are clearly monotonic in : in the absence of redistribution, each individual prefers the tax rate that fully corrects the distortion in consumption. Since the distortion term (1 ) h0(x (q; )) is decreasing in , the individually preferred tax rate is monotonically increasing in the level of self-control problems. Given this monotonicity, the tax rate chosen in a majority voting equilibrium is
1 8In the present setting, it would also be possible to set individual-speci…c taxes. However, this case would not be very interesting, as there would then be no interpersonal trade-o¤s to be settled. The case of individual- speci…c transfers is also rather unrealistic, but it is useful for illustrating some of the key mechanisms in this paper, and serves as a benchmark for the more realistic case where sin taxes have redistributive e¤ects.
1 9It is important to note thatS( ; )is a lump-sum payment, and the consumer cannot change the subsidy by deviating fromx :
2 0d2V(q; i)
dq2 = [v00(x ) h00(x )] dxdq
2
[v0(x ) h0(x ) 1]ddq2x2 <0:
given by
= (1 med) h0(x (q; med)): (9)
We are now in a position to compare the tax rate chosen in political equilibrium, ; to the socially optimal tax rate o. Let us …rst examine a simple case, where the harm function is linear, namelyh(x) =gx:In this case = g(1 med):On the other hand, equation (5) implies that o = g(1 E[ ] + ); where Covh
(1 );@x@q(q; )i
=E h@x (q; )
@q
i
> 0:
Then clearly < o;as long as E[ ]< med+ . That is, the equilibrium tax rate is lower than the socially optimal rate, as long as there are enough relatively rational individuals (when the distribution of is not too much skewed to the right). In the appendix, we show that this result holds also for more general harm functions.
Proposition 2 Assume that sin taxes have no redistributive e¤ ects and the optimal level of consumption at zero taxes is positive. If the distribution of is not too much skewed to the right, the socially optimal tax rate is higher than the tax rate chosen in a majority voting equilibrium:
Proof. See the appendix.
The reader may worry that since the tax formulae in (5) and (9) are given only in implicit form, the result in Proposition 2 might only be related to taxrules, and not to actual tax levels. Limitations of this type are very common in the optimal taxation literature (see for example Gaube (2005)). However, using a Taylor-approximation of (7), we have further shown in the appendix that dWdq(q) @V(q;@qmed) 0 for all q, and therefore the result that
o > is robust to this objection.21 This conclusion holds as long as the approximation used in the proof can be considered to be valid. Given that we are in the current subsection concerned with goods that are moderately harmful (and tax rates should therefore not be very high), the approximation is likely to be fairly innocuous.
The case of a symmetric distribution of is worth emphasising - in this case, intuition might suggest that the equilibrium and the socially optimal tax rates should coincide, but we have shown that the result o > nevertheless holds: there is a kind of bias in voting behaviour that tends to make the equilibrium tax rate too low.
In order to further clarify the intuition behind the result stated in Proposition 2, let us examine a speci…c example where there are three individuals with 1 = 0; 2 = 12 and 3= 1.
Then med=E[ ] = 12. The tax rate chosen in political equilibrium, now equals the tax
2 1To show that dWdq(q) @V(q;@qmed) >0implies qo > q ;denote (q) dWdq(q) @V(q;@qmed) >0and e W(1) V (1; med). NowW(q) =V (q; med) +Rq
1 (bq)dqb+e:The price level chosen in the political equilib- rium isq = arg maxqV (q; med). Next we show that the socially optimal price levelqo= arg maxW(q)> q . (i) Assume by contrast thatqo=q < qe . NowW(q) =e W(q ) [V(q ; med) V(eq; med)] Rq
e
q (bq)dq <b W(q ): Thus eq cannot be optimal, a contradiction. (ii) W0(q ) = @V(q ; med)
@q + (q ) > 0, where the inequality follows since @V(q ; med)
@q = 0and (q )>0. Thus we can conclude thatqo> q . Notice that this proof applies even ifW(q) andV(q)are multi-peaked.
rate preferred by individual 2. The tax rate is socially optimal if and only if dW(q )
dq
@V (q ; 1)
@q +@V (q ; 2)
@q +@V (q ; 3)
@q = 0
Because dV(qdq; 2) = 0;this condition can be written as
@V (q ; 1)
@q = @V (q ; 3)
@q (10)
Typically, the condition (10) does not hold in equilibrium. As an example, assume that v(x) = ""1x""1 andh(x) =gxand the demand function is thereforex (q; ) = (q+ ) ": With these functional forms, the tax rate chosen in political equilibrium is given by
@V (q ; 2)
@q = 0, = (1 2) g= 1 2 g:
We then have that
@V (q ; 1)
@q = [ (1 1) g]@x (q ; 1)
@q = 1
2 g" 1 +1 2
("+1)
>0
@V (q ; 3)
@q = [ (1 3) g]@x (q ; 3)
@q = 1
2 g" 1 +3 2
("+1)
<0
For individual 1 with a very severe self-control problem, the best outcome would be to have 1 = g;the preferred outcome of the fully rational individual 3 would be to have no taxes, 3 = 0:In other words, the equilibrium tax rate is too low from the point of view of consumer 1 and too high from the point of view of consumer 3. Further, the absolute value of the di¤erence between the equilibrium tax rate and each individual’s preferred tax rate is the same for both individuals 1 and 3:22
j ij= j (1 i) gj= 1
2 g"; i= 1;3:
Nevertheless, we have that
@x (q ; 1)
@q = 1 +1
2 g
("+1)
> 1 +3 2 g
("+1)
= @x (q ; 3)
@q :
In other words, a change in the tax rate a¤ects the consumption decision of individual 1 more than it a¤ects the consumption choice of individual 3. The result is very intuitive: the rational individual has a low level of consumption, and therefore increasing the sin tax cannot have a very large e¤ect on his consumption level (in absolute terms). On the other hand, the individual with a severe self-control problem has a high level of consumption, and a tax
2 2This equality is caused by the linearity ofh(x).
increase has a larger e¤ect on his level of consumption. Therefore we have that
@V (q ; 1)
@q > @V (q ; 3)
@q and thus
dW(q ) dq >0:
This example con…rms our intuition that if we take the political equilibrium as a starting point and increase the tax rate slightly, the self-control bene…t gained by the individual with a severe self-control problem is higher than the loss experienced by close to rational individuals.
However, the median voter does not take this asymmetric e¤ect into account. Therefore even if the distribution of self-control problems is symmetric, the equilibrium tax rate is below the socially optimal level.
4.1.2 The case of very harmful consumption
Consider next the case where consumption of commodity x is so harmful that the optimal (rational) level of consumption is zero even at zero taxes, that is, v0(0) h0(0) 1 0.
It is then immediately clear that, in the social optimum, no one should consume x: The (minimum) tax rate( o) needed to implement the social optimum is such that even the least rational consumer abstains, and it is given by
o =v0(0) L h0(0) 1: (11)
It is easy to see that the socially optimal sin tax is in this case also a majority voting equilibrium for any distribution of and for all functional formsv(x)and h(x):
Proposition 3 Assume that sin taxes have no redistributive e¤ ects and the optimal level of consumption at zero taxes is zero. The socially optimal tax is then a majority voting equilibrium:
Proof. Individuals with i= Lstrictly prefer oto any other tax rate. All individuals with
i> L strictly prefer o to any tax rate below^ = v0(0) i h0(0) 1 and are indi¤erent between o and any tax rate [^; o]:
That is, when it is optimal to abstain from the consumption of goodxeven in the absence of any taxation, all consumers prefer a tax policy that will help them to achieve a zero level of consumption. The socially optimal tax achieves this outcome and will therefore be a majority voting equilibrium. However, we show in section 4.2.2 that this result changes when we take into account the redistributive e¤ects of sin taxes.
4.2 Accounting for the redistributive e¤ects of sin taxes
Let us next analyse the more realistic case where the social planner does not have information on individual consumption levels, so that the subsidy paid to each consumer cannot be conditioned on individual consumption. We therefore assume from now on that all consumers receive a lump-sum transfer of equal size, and this subsidy is given by S(q; ) = S(q) =
E [x(q; )]:The consumers’policy preference function is then given by V~ (q; i) =v(x (q; i)) h(x (q; i)) qx + E [x (q; )] +B:
4.2.1 The case of moderately harmful consumption
Consider again …rst the case where current consumption causes harm in the future, but the optimal (rational) level of consumption is nevertheless positive at zero taxes, that is, v0(0) h0(0) 1>0:The …rst-order condition that determines voting behaviour of individual iis now given by
@V~(q; i)
@q = (1 i) h0(x (q; i))@x (q; i)
@q (12)
x (q; i) +E [x (q; )] + E @x (q; )
@q :
In the case where taxation had no redistributive e¤ects it was easy to see that the indi- vidual’s preferred tax rate was monotonic in the level of self-control problems and a majority voting equilibrium was therefore guaranteed to exists. However, as noted in the introduction, in the case where sin taxes have redistributive e¤ects there are two forces at play: on the one hand, a person with a high level of self-control problems will prefer a high tax in order to alleviate the distortion in his consumption decision. The corrective e¤ect of taxation is iden- tical to the case where sin taxes have no redistributive e¤ects, and is given by the …rst term in (12). On the other hand, however, a high tax will also imply a transfer of income towards individuals with a relatively low level of consumption: this redistributive e¤ect of taxation is captured by the remaining terms in (12). Because of these two opposite e¤ects, policy preferences may not be well-behaved, and the existence of a majority voting equilibrium is therefore not self-evident in this case.
A median voter equilibrium exists if policy preferences satisfy the Gans-Smart single crossing property (Gans and Smart 1996). Gans and Smart show that when underlying preferences are de…ned over a two-dimensional real choice variable but attention can be restricted to a one-dimensional choice due to production or budget constraints (in our case due to the consumer’s budget constraint), then single-crossing in the Spence-Mirrlees sense implies single-crossing in the Gans-Smart sense.
We therefore use the Spence-Mirrlees single-crossing condition to analyse the existence of
a median voter equilibrium. The voters’preferences satisfy the Spence-Mirrlees condition if voters’marginal rates of substitution betweenz andq; V~~q
Vz;are globally monotonic in . In our simple case of quasilinear preferences (V~z = 1), the condition reduces to @V~(q;@q i) being monotonic in :
In the appendix, we prove that @V~@q@(q; i) 0, and we can therefore state the following proposition:
Proposition 4 Assume that revenue from sin taxes is distributed equally among consumers.
A majority voting equilibrium exists and the equilibrium sin tax is given by the tax rate preferred by the consumer with the median level of self-control problems.
Proof. See the appendix.
The result @V~@q@(q; i) 0 implies that an individuals’ most preferred tax rate is again monotonically increasing in the level of self-control problems. To gain some intuition on why this holds also when the redistributive e¤ects of sin taxes are taken into account, using equations (16) and (23) (see the appendix), the corrective term in (12) can be written as
(1 i) h0(x (q; i))@x (q; i)
@q =x (q; i) x (q; 1) + (q; i); (13) where (q; i) = R H
i (1 ^)@2x(q;^)
@b2 db > 0 and (q; i) is increasing in the level of self- control problems (decreasing in ). As expected, the corrective e¤ect of the tax is the larger, the more consumption x (q; i) di¤ers from the quantity chosen by the rational consumer, x (q; 1). Further, the magnitude of the corrective e¤ect exceeds the di¤erence x (q; i) x (q; 1). Importantly, the di¤erence between the corrective e¤ect and the monetary cost of the tax is increasing in the level of self-control problems: therefore, relatively irrational individuals prefer a higher tax rate than those who are relatively rational.
To illustrate, consider again the example where v(x) = ""1x""1 and h(x) =gx: With these functional forms, the self-control bene…t from taxation can be expressed as
(1 i) h0(x (q; i))@x (q; i)
@q ="( g+q)x
"+1
"
i "xi:
It is easy to see from this expression that the self-control bene…t increases more than linearly with the quantity consumed, which in turn is an increasing and convex function of self-control problems. Hence the self-control bene…ts increase more rapidly than monetary costs (which depend linearly on consumption), and the individuals’preferred tax rate is increasing in the level of self-control problems.
The speci…c functional forms used in the previous example serve to illustrate the mech- anism behind Proposition 4. However, the property that self-control bene…ts increase more rapidly than monetary costs is more general, as Proposition 4 holds for all functional forms
that satisfy Assumption 1. The intuition is apparent when we examine the three compo- nents that a¤ect the magnitude of self-control bene…ts: Firstly, the harm function has been assumed to be either linear or convex. Secondly, the sensitivity of demand to tax changes increases with self-control problems. Finally, these two e¤ects are multiplied by the level of self-control problems, (1 i). Hence, these three forces reinforce each other, causing the self-control bene…ts to quite generally increase more than linearly in the level of consumption.
It is interesting to note that even in the present case where the optimal level of con- sumption in the absence of taxes is positive, there may be circumstances under which the redistributive motive for taxation implies that the median voter prefers not to consume in equilibrium, and will vote for the tax rate that maximises revenue. We have shown in the proof of Proposition 4, however, that single-crossing holds regardless of whether the median voter consumes the good in equilibrium or not. Therefore there cannot be situations where a coalition of near-rational users and highly irrational heavy-users vote for highest taxes, and are pitted against voters with an intermediate degree of rationality23.
Let us next turn to the comparison between the equilibrium and the optimum. Proposition 4 implies that the equilibrium tax rate is now given by
=
(1 med) h0(x (q; med)) @x (q;@qmed) x (q; med) +E [x (q; )]
E @x@q(q; )
:
The termE [x (q; )] x (q; med)in the above formula captures the fact that ifx (q; med)<
E [x (q; )], sin taxes imply a transfer of income towards the median voter. This will typi- cally occur if the distribution of is skewed to the left so that med> E[ ](though it can also occur in other cases, depending on the exact functional form ofx ). In such circumstances, the median voter then votes for a higher tax than he would in the absence of redistribution.
Nevertheless, we can show that the equilibrium tax rate is again typically below the socially optimal level:
Proposition 5 Assume that revenue from sin taxes is distributed equally among consumers.
If (1 )@2@x(q; )2 is non-increasing in and the distribution of is not too much skewed to the right, the socially optimal tax rate is higher than the tax rate chosen in a majority voting equilibrium:
Proof. See the appendix.
2 3It appears that this property might be related to our assumption of quasilinear preferences, and the implied zero income elasticity of demand for goodx. Whether more general assumptions about preferences can give rise to voting coalitions where highly irrational consumers and fully rational consumers vote for higher taxes than consumers with an intermediate level of rationality, is left as a question for further research. See Epple and Romano (1996) for an analysis - albeit in a very di¤erent context - where the preferred level of public expenditure and taxation is non-monotonic in consumer type (in their case income) when the income elasticity of demand exceeds the (absolute value of the) price elasticity.
In the appendix, we show that if(1 )@2x(q; )@ 2 is non-increasing in , the derivative (12) is not only increasing, but also a convex function of the level of self-control problems. The result therefore again has a very intuitive explanation: the convexity of (12) implies that the marginal welfare bene…t of high taxation for relatively irrational individuals is higher than the corresponding welfare loss for close to rational individuals. The condition that(1 )@2x(q; )
@ 2
should be non-increasing in again holds for many commonly used functional forms, for example whenv is of the CRRA or CARA-variety or quadratic, and when the harm function is linear orh(x) =xs where s 2.
Further, in order to interpret the condition that (1 )@2x(q; )@ 2 should be non-increasing in , we show in the appendix that this holds (approximately) if a price change a¤ects the health of irrational consumers (heavy users) more than the health of rational consumers.
This holds given Assumption 1.
Proposition 5 shows that despite the fact that high sin taxes result in a transfer of income towards the median voter, the equilibrium tax rate is lower than the socially optimal tax:
the income transfer is not su¢ cient for the median voter to fully internalise the bene…t that would accrue to highly irrational individuals, if the tax rate was increased.
For the case of moderately harmful consumption, therefore, our result is identical to the benchmark case where sin taxes have no redistributive e¤ects. If harm from consumption is very low, this result is intuitive: in this case even rational consumers consume a lot of the good in question, and the concern for alleviating distortions in consumption is stronger than the motive for redistributing tax revenue. This case therefore resembles the case where taxation has no redistributive e¤ects, discussed above, and the results are the same for the two cases - equilibrium taxes tend to be too low.
However, in contrast to the case where sin taxes have no redistributive e¤ects, it is important to note that the conclusion that equilibrium taxes tend to be too low holds in the present case forall levels of harm. We turn next to the interesting case of very harmful consumption.
4.2.2 The case of very harmful consumption
Consider again the case where consumption is so harmful that at zero taxes, the optimal choice is to abstain from consuming the commodityx. The (minimum) tax rate( o)needed to implement the social optimum is then given by (11).
The condition for the result < o to hold in this case is especially mild, namely that
med does not coincide with the lowest level of . Further it is important to note that this result does not depend on Assumption 1, but holds forany functional forms ofv(x)andh(x):
Proposition 6 Assume that revenue from sin taxes is distributed equally among consumers and consumption is so harmful that the optimal level of consumption at zero taxes is zero.
Then, for any distribution of self-control problems where med> L, the socially optimal tax rate is higher than the tax rate chosen in a majority voting equilibrium.
Proof. Since no agent consumes at = o, no tax revenues are collected in the social optimum. Suppose that, starting from = o, the tax rate is lowered by a small amountd . Then the least rational consumers ( L);who were just indi¤erent between consuming and not consuming, are triggered to consume a small amount
x( L; o d ) = d
v00(0) + Lh00(0) >0:
Also, tax revenues increase from zero to v00(0)+od
Lh00(0). Consumers other than type L still consume no x, and the welfare of this majority group increases, due to transfers from type
L. The redistributive motive for taxation then implies that the level of taxes that eliminates all consumption cannot arise as a political equilibrium.
5 Extension: third-best sin taxes
Our analysis can be easily generalised to the case where sin taxes not only have a corrective role, but the government also has a revenue raising objective. Total social welfare is now taken to be the sum of the utility from private consumption and from public funds. Let denote the marginal social value of revenue from the sin tax (or the marginal cost of raising revenue from other tax bases)24. Previously, we have had that = 1, since all tax revenue has been returned to consumers and we have assumed that the marginal utility of (private) income is 1. However, now we allow for the possibility that part of the tax revenue is used for public goods and services. In the general case where the marginal value of public consumption di¤ers from the marginal value of private consumption, we have that 6= 1.
Social welfare is now given by
W (q; ) =E [V (q; ; )] =E [v(x (q; )) h(x (q; )) (1 + (1 ) )x (q; )] +B:
This extension does not a¤ect the comparison between the equilibrium and the socially opti- mal tax rate25. To see this, note that the individual’s policy preference function is now given by
V~ (q; i) =v(x (q; i)) h(x (q; i)) qx + E [x (q; )] +B
2 4For simplicity, we consider a partial equilibrium set-up where demands for other goods are assumed to be independent of the demand for the good under consideration and consequently, other tax bases are una¤ected by the taxes in question. It is therefore also natural to assume that is constant with respect to .
2 5This holds as long as we make the conventional assumption that consumers’ and the social planner’s valuations of public funds are identical. See Pirttilä and Tenhunen (2007) for an analysis where those valuations may di¤er.
and its derivative is
@V~(q; i)
@q = (1 i) h0(x (q; i))@x (q; i)
@q
x (q; i) + E [x (q; )] + E @x (q; )
@q :
It is important to note that the part of this derivative that depends on is independent of , and therefore @V~(q;@q i) is still increasing and convex in the level of self-control problems, as above. The equilibrium tax rate is therefore typically lower than the socially optimal tax also in this third-best setting. We can therefore state the following proposition:
Proposition 7 Assume that sin taxes are used for revenue raising purposes. If(1 )@2x(q; )
@ 2
is non-increasing in and the distribution of is not too much skewed to the right, the third- best optimal sin tax is higher than the tax rate chosen in a majority voting equilibrium:
However, even though the qualitative comparison between the socially optimal and the equilibrium tax rate is not altered, incorporating a revenue raising objective into the social welfare function does a¤ect the actual levels of taxes. Consider the socially optimal tax rate.
The third-best optimal sin tax is given by
T B= 1E h
(1 ) h0(x (q; ))@x@q(q; )i E h@x (q; )
@q
i +(1 )E [x (q; )]
E h@x (q; )
@q
i: (14)
The …rst term in (14) re‡ects the corrective role of sin taxes, whereas the second term re‡ects the standard public …nance argument for taxation. The corrective part enters the optimal tax formula additively, in accordance with the additivity principle familiar from the context of environmental taxation (Sandmo 1975). If = 1, we get the formula in (5).
Typically, however, if other taxes are distortionary, > 1 and the public …nance term is positive. The direct e¤ect of public …nance considerations is therefore to increase the socially optimal sin tax.
On the other hand, for > 1 the corrective part of the tax is scaled down. The fact that the corrective part is lower than in the case where taxation only has a corrective role is analogous to the principle of incomplete internalisation of environmental externalities in the presence of distortive taxation (Bovenberg and de Mooij 1994). This feature can be explained by the fact that the marginal costs of harm reduction increase with the marginal cost of public funds.
The overall e¤ect of public …nance considerations on the magnitude of the optimal sin tax is in general ambiguous. However, it is self-evident that in the case where >1;that is when tax revenue has a higher weight in the social welfare function than private consumption, the
third-best optimal sin tax is higher than the tax rate given in (5) if and only if (5) is on an upward-sloping segment of the La¤er curve.
In some cases, for example if a part of tax revenue is wasted (for example due to admin- istrative costs) a case where <1 may be relevant. In this case, the second term in (14) is negative whereas the …rst part is scaled up. Again, it is worth emphasising that our result regarding the comparison between and o does not depend on the value of . Therefore, the result that the optimal tax rate exceeds the equilibrium level of taxes holds even in the case where part of tax revenue is wasted.
6 Conclusions
We have shown that optimal sin taxes will typically exceed the average distortion caused by self-control problems in the economy: this result arises due to the asymmetric e¤ects of sin taxes on the welfare of those with severe self-control problems on the one hand, and on (close to) rational individuals on the other hand. As the median voter does not take such asymmetries into account, the sin taxes chosen in a majority voting equilibrium are in most cases below the socially optimal level.
The view that emerges from previous empirical literature seems to be that for example excise duties on cigarettes are in most countries very high compared to the external costs of smoking. However, the present analysis provides a theoretical argument that suggests that such taxes may nevertheless be too low from a social point of view.
The analysis also suggests a reason for why it may be optimal to impose quantity restric- tions on the consumption of certain highly harmful substances (such as illicit drugs), rather than using price instruments alone: if we were to rely solely on tax policy to regulate the consumption of such substances, the optimal outcome may not be reached.
It should be noted that throughout the analysis, we have assumed that individuals are sophisticated - that is, they are fully aware of their self-control problem. Individuals thus value sin taxes as a self-control device, and vote for positive taxes. However, if some individuals are partially naive, their preferred tax rate will be lower than the level indicated by our results - indeed, fully naive individuals would vote for a zero tax, as they are fully unaware of their self-control problem. In the case where some individuals are either partially or fully naive, therefore, the problem of sub-optimally low equilibrium taxes would be exacerbated.
Appendix
Preliminaries
The following derivatives are used a number of times in the analysis:
@x (q; )
@ = h0(x )
v00(x ) h00(x ) <0 (15)
@x (q; )
@q = 1
v00(x ) h00(x ) <0:
Given these results, the corrective e¤ect of taxation can be written as (1 i) h0(x (q; i))@x (q; i)
@q = (1 i)@x(q; )
@ : (16)
Proof of Proposition 1 SinceE h
@x
@q
i
<0, o> E [(1 )h0(x (q; ))] ifCov h
(1 )h0(x );@x@q i
<0. When h00(x) 0,(1 )h0(x (q; )) is decreasing in :Therefore, Cov h
(1 )h0(x );dxdq i
<0 if @2@q@x (q; ) >0. This derivative is given by
@2x (q; )
@q@ = [v000(x ) h000(x )]@x@ + h00(x ) [v00(x ) h00(x )]2 : It can be shown that @2x@q@(q; ) >0 if (su¢ cient conditions)v000(x) 0and
h000(x)h0(x)
[h00(x)]2 < h00(x) v00(x) h00(x) :
Clearly h00(x)h00(x)v00(x) >1. Thus the above condition is less demanding than h000(x)h0(x)
[h00(x)]2 1:
Proof of Proposition 2
In this proof, we …rst show that the right hand side of (5) is larger than the right hand side of (9). Secondly, we derive a condition that guarantees dW(q)dq @V(q;@qmed) 0. Thirdly, we interpret this condition, and use an approximation to show that it holds in our model.
i) In the text we show that the proposition holds for a linear h(x). Ifh(x) is not linear, then
E (1 )h0(x (q; )) 6= (1 med)h0(x (q; med))
even if med=E[ ]. In particular, if ( ) = (1 )h0(x (q; )) is a convex function of , then the Jensen inequality implies that
E (1 )h0(x (q; )) >(1 E[ ])h0(x (q;E[ ])) = (1 med)h0(x (q; med)):
